Investigators looking into last year’s data breach and theft of drivers’ records from Uber Technologies Inc. have found indications implicating an executive at rival ride-hailing app Lyft Inc., said people familiar with the matter.

The Uber investigators believe the intruder used a company security key accessible on a public website to access as many as 50,000 driver records in May 2014, Uber said in court records in a lawsuit related to the breach. Uber discovered the breach in September 2014 and disclosed it in February....